Difference between revisions of "2025 AIME I Problems"
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+ | {{AIME Problems|year=2025|n=I}} | ||
+ | ==Problem 1== | ||
+ | |||
+ | Find the sum of all integer bases <math>b > 9</math> for which <math>17_b</math> is a divisor of <math>97_b</math>. | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 1|Solution]] | ||
+ | |||
+ | ==Problem 2== | ||
+ | |||
+ | On <math>\triangle ABC</math> points <math>D</math> and <math>E</math> lie on <math>\overline{AB}</math> so that <math>AD < AE < AB</math>, while points <math>F</math> and <math>G</math> lie on <math>\overline{AC}</math> so that <math>AF < AG < AC</math>. Suppose <math>AD = 4</math>, <math>DE = 16</math>, <math>EB = 8</math>, <math>AF = 13</math>, <math>FG = 52</math>, and <math>GC = 26</math>. Let <math>M</math> be the reflection of <math>D</math> through <math>F</math>, and let <math>N</math> be the reflection of <math>G</math> through <math>E</math>. The area of quadrilateral <math>DEGF</math> is <math>288</math>. Find the area of heptagon <math>AFNBCEM</math>, as shown in the figure below. | ||
+ | |||
+ | <asy> | ||
+ | unitsize(14); | ||
+ | pair A = (0, 9), B = (-6, 0), C = (12, 0), D = (5A + 2B)/7, E = (2A + 5B)/7, F = (5A + 2C)/7, G = (2A + 5C)/7, M = 2F - D, N = 2E - G; | ||
+ | filldraw(A--F--N--B--C--E--M--cycle, lightgray); | ||
+ | draw(A--B--C--cycle); | ||
+ | draw(D--M); | ||
+ | draw(N--G); | ||
+ | dot(A); | ||
+ | dot(B); | ||
+ | dot(C); | ||
+ | dot(D); | ||
+ | dot(E); | ||
+ | dot(F); | ||
+ | dot(G); | ||
+ | dot(M); | ||
+ | dot(N); | ||
+ | label("$A$", A, dir(90)); | ||
+ | label("$B$", B, dir(225)); | ||
+ | label("$C$", C, dir(315)); | ||
+ | label("$D$", D, dir(135)); | ||
+ | label("$E$", E, dir(135)); | ||
+ | label("$F$", F, dir(45)); | ||
+ | label("$G$", G, dir(45)); | ||
+ | label("$M$", M, dir(45)); | ||
+ | label("$N$", N, dir(135)); | ||
+ | </asy> | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 2|Solution]] | ||
+ | |||
+ | ==Problem 3== | ||
+ | |||
+ | The <math>9</math> members of a baseball team went to an ice-cream parlor after their game. Each player had a single scoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let <math>N</math> be the number of different assignments of flavors to players that meet these conditions. Find the remainder when <math>N</math> is divided by <math>1000.</math> | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 3|Solution]] | ||
+ | |||
+ | ==Problem 4== | ||
+ | Find the number of ordered pairs <math>(x,y)</math>, where both <math>x</math> and <math>y</math> are integers between <math>-100</math> and <math>100</math> inclusive, such that <math>12x^2-xy-6y^2=0</math>. | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 4|Solution]] | ||
+ | |||
+ | ==Problem 5== | ||
+ | |||
+ | There are <math>8!= 40320</math> eight-digit positive integers that use each of the digits <math>1, 2, 3, 4, 5, 6, 7, 8</math> exactly once. Let <math>N</math> be the number of these integers that are divisible by <math>22</math>. Find the difference between <math>N</math> and <math>2025</math>. | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 5|Solution]] | ||
+ | |||
+ | ==Problem 6== | ||
+ | |||
+ | An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is 3, and the area of the trapezoid is 72. Let the parallel sides of the trapezoid have lengths <math>r</math> and <math>s</math>, with <math>r \neq s</math>. Find <math>r^2 + s^2</math>. | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 6|Solution]] | ||
+ | |||
+ | ==Problem 7== | ||
+ | |||
+ | The twelve letters <math>A</math>,<math>B</math>,<math>C</math>,<math>D</math>,<math>E</math>,<math>F</math>,<math>G</math>,<math>H</math>,<math>I</math>,<math>J</math>,<math>K</math>, and <math>L</math> are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is <math>AB</math>, <math>CJ</math>, <math>DG</math>, <math>EK</math>, <math>FL</math>, <math>HI</math>. The probability that the last word listed contains <math>G</math> is <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 7|Solution]] | ||
+ | |||
+ | ==Problem 8== | ||
+ | |||
+ | Let <math>k</math> be a real number such that the system | ||
+ | \begin{align*} | ||
+ | &|25 + 20i - z| = 5 \ | ||
+ | &|z - 4 - k| = |z - 3i - k| | ||
+ | \end{align*} | ||
+ | has exactly one complex solution <math>z</math>. The sum of all possible values of <math>k</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. Here <math>i = \sqrt{-1}</math>. | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 8|Solution]] | ||
+ | |||
+ | ==Problem 9== | ||
+ | |||
+ | The parabola with equation <math>y = x^2 - 4</math> is rotated <math>60^{\circ}</math> counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has <math>y</math>-coordinate <math>\frac{a - \sqrt{b}}{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, and <math>a</math> and <math>c</math> are relatively prime. Find <math>a + b + c</math>. | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 9|Solution]] | ||
+ | |||
+ | ==Problem 10== | ||
+ | |||
+ | The <math>27</math> cells of a <math>3 \times 9</math> grid are filled in using the numbers <math>1</math> through <math>9</math> so that each row contains <math>9</math> different numbers, and each of the three <math>3 \times 3</math> blocks heavily outlined in the example below contains <math>9</math> different numbers, as in the first three rows of a Sudoku puzzle. | ||
+ | |||
+ | <asy> | ||
+ | unitsize(20); | ||
+ | |||
+ | add(grid(9,3)); | ||
+ | |||
+ | draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2)); | ||
+ | draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2)); | ||
+ | |||
+ | real a = 0.5; | ||
+ | |||
+ | label("5",(a,a)); | ||
+ | label("6",(1+a,a)); | ||
+ | label("1",(2+a,a)); | ||
+ | label("8",(3+a,a)); | ||
+ | label("4",(4+a,a)); | ||
+ | label("7",(5+a,a)); | ||
+ | label("9",(6+a,a)); | ||
+ | label("2",(7+a,a)); | ||
+ | label("3",(8+a,a)); | ||
+ | |||
+ | label("3",(a,1+a)); | ||
+ | label("7",(1+a,1+a)); | ||
+ | label("9",(2+a,1+a)); | ||
+ | label("5",(3+a,1+a)); | ||
+ | label("2",(4+a,1+a)); | ||
+ | label("1",(5+a,1+a)); | ||
+ | label("6",(6+a,1+a)); | ||
+ | label("8",(7+a,1+a)); | ||
+ | label("4",(8+a,1+a)); | ||
+ | |||
+ | label("4",(a,2+a)); | ||
+ | label("2",(1+a,2+a)); | ||
+ | label("8",(2+a,2+a)); | ||
+ | label("9",(3+a,2+a)); | ||
+ | label("6",(4+a,2+a)); | ||
+ | label("3",(5+a,2+a)); | ||
+ | label("1",(6+a,2+a)); | ||
+ | label("7",(7+a,2+a)); | ||
+ | label("5",(8+a,2+a)); | ||
+ | |||
+ | </asy> | ||
+ | |||
+ | The number of different ways to fill such a grid can be written as <math>p^a \cdot q^b \cdot r^c \cdot s^d</math> where <math>p</math>, <math>q</math>, <math>r</math>, and <math>s</math> are distinct prime numbers and <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> are positive integers. Find <math>p \cdot a + q \cdot b + r \cdot c + s \cdot d</math>. | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 10|Solution]] | ||
+ | |||
+ | ==Problem 11== | ||
+ | A piecewise linear function is defined by <cmath>f(x) = | ||
+ | |||
+ | The parabola <math>x = 34y^{2}</math> intersects the graph of <math>f(x)</math> at finitely many points. The sum of the <math>y</math>-coordinates of all these intersection points can be expressed in the form <math>\tfrac{a + b\sqrt{c}}{d}</math>, where <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are positive integers such that <math>a</math>, <math>b</math>, <math>d</math> have greatest common divisor equal to <math>1</math>, and <math>c</math> is not divisible by the square of any prime. Find <math>a + b + c + d</math>. | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 11|Solution]] | ||
+ | |||
+ | ==Problem 12== | ||
+ | |||
+ | The set of points in <math>3</math>-dimensional coordinate space that lie in the plane <math>x+y+z=75</math> whose coordinates satisfy the inequalities <cmath>x-yz<y-zx<z-xy</cmath>forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form <math>a\sqrt{b},</math> where <math>a</math> and <math>b</math> are positive integers and <math>b</math> is not divisible by the square of any prime. Find <math>a+b.</math> | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 12|Solution]] | ||
+ | |||
+ | ==Problem 13== | ||
+ | |||
+ | Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws <math>25</math> more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these <math>27</math> line segments divide the disk. | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 13|Solution]] | ||
+ | |||
+ | ==Problem 14== | ||
+ | |||
+ | Let <math>ABCDE</math> be a convex pentagon with <math>AB=14,</math> <math>BC=7,</math> <math>CD=24,</math> <math>DE=13,</math> <math>EA=26,</math> and <math>\angle B=\angle E=60^{\circ}.</math> For each point <math>X</math> in the plane, define <math>f(X)=AX+BX+CX+DX+EX.</math> The least possible value of <math>f(X)</math> can be expressed as <math>m+n\sqrt{p},</math> where <math>m</math> and <math>n</math> are positive integers and <math>p</math> is not divisible by the square of any prime. Find <math>m+n+p.</math> | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 14|Solution]] | ||
+ | |||
+ | ==Problem 15== | ||
+ | |||
+ | Let <math>N</math> denote the number of ordered triples of positive integers <math>(a, b, c)</math> such that <math>a, b, c \leq 3^6</math> and <math>a^3 + b^3 + c^3</math> is a multiple of <math>3^7</math>. Find the remainder when <math>N</math> is divided by <math>1000</math>. | ||
+ | |||
+ | [[2025 AIME I Problems/Problem 15|Solution]] | ||
+ | |||
+ | ==See also== | ||
+ | {{AIME box|year=2025|n=I|before=[[2024 AIME II Problems|2024 AIME II]]|after=[[2025 AIME II Problems|2025 AIME II]]}} | ||
+ | * [[American Invitational Mathematics Examination]] | ||
+ | * [[AIME Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 02:12, 18 February 2025
2025 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
[hide]Problem 1
Find the sum of all integer bases for which
is a divisor of
.
Problem 2
On points
and
lie on
so that
, while points
and
lie on
so that
. Suppose
,
,
,
,
, and
. Let
be the reflection of
through
, and let
be the reflection of
through
. The area of quadrilateral
is
. Find the area of heptagon
, as shown in the figure below.
Problem 3
The members of a baseball team went to an ice-cream parlor after their game. Each player had a single scoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let
be the number of different assignments of flavors to players that meet these conditions. Find the remainder when
is divided by
Problem 4
Find the number of ordered pairs , where both
and
are integers between
and
inclusive, such that
.
Problem 5
There are eight-digit positive integers that use each of the digits
exactly once. Let
be the number of these integers that are divisible by
. Find the difference between
and
.
Problem 6
An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is 3, and the area of the trapezoid is 72. Let the parallel sides of the trapezoid have lengths and
, with
. Find
.
Problem 7
The twelve letters ,
,
,
,
,
,
,
,
,
,
, and
are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is
,
,
,
,
,
. The probability that the last word listed contains
is
, where
and
are relatively prime positive integers. Find
.
Problem 8
Let be a real number such that the system
. The sum of all possible values of
can be written as
, where
and
are relatively prime positive integers. Find
. Here
.
Problem 9
The parabola with equation is rotated
counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has
-coordinate
, where
,
, and
are positive integers, and
and
are relatively prime. Find
.
Problem 10
The cells of a
grid are filled in using the numbers
through
so that each row contains
different numbers, and each of the three
blocks heavily outlined in the example below contains
different numbers, as in the first three rows of a Sudoku puzzle.
The number of different ways to fill such a grid can be written as where
,
,
, and
are distinct prime numbers and
,
,
,
are positive integers. Find
.
Problem 11
A piecewise linear function is defined by and
for all real numbers
. The graph of
has a sawtooth pattern.
The parabola intersects the graph of
at finitely many points. The sum of the
-coordinates of all these intersection points can be expressed in the form
, where
,
,
, and
are positive integers such that
,
,
have greatest common divisor equal to
, and
is not divisible by the square of any prime. Find
.
Problem 12
The set of points in -dimensional coordinate space that lie in the plane
whose coordinates satisfy the inequalities
forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form
where
and
are positive integers and
is not divisible by the square of any prime. Find
Problem 13
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these
line segments divide the disk.
Problem 14
Let be a convex pentagon with
and
For each point
in the plane, define
The least possible value of
can be expressed as
where
and
are positive integers and
is not divisible by the square of any prime. Find
Problem 15
Let denote the number of ordered triples of positive integers
such that
and
is a multiple of
. Find the remainder when
is divided by
.
See also
2025 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2024 AIME II |
Followed by 2025 AIME II | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.